X = K under review

TL;DR

This paper reviews the X = K conjecture, discusses proof components, and introduces an algorithm for calculating the combinatorial R-matrix and coenergy function, highlighting a surprising relation valid at small ranks.

Contribution

It provides a comprehensive review of the X = K conjecture, explores rank estimates, and presents a novel algorithm for key calculations in the theory.

Findings

The X = K conjecture is reviewed with key proof ingredients.

A new algorithm for the combinatorial R-matrix and coenergy function is introduced.

A strange relation is identified that holds even at small ranks.

Abstract

We review the X = K conjecture and important ingredients for the proof. We also attach notes on the rank estimate for the X = K theorem to hold and on the strange relation that was found to be valid without the assumption that the rank is sufficiently large. Using the latter one obtains an algorithm to calculate the image of the combinatorial R-matrix and the value of the coenergy function.